Zero distribution for Angelesco Hermite–Padé polynomials

Abstract

We consider the problem of zero distribution of the first kind Hermite-Padé polynomials associated with a vector function f = (f 1 , . . . , f s ) whose components f k are functions with a finite number of branch points in plane. We assume that branch sets of component functions are well enough separated (which constitute the Angelesco case). Under this condition we prove a theorem on limit zero distribution for such polynomials. The limit measures are defined in terms of a known vector equilibrium problem.Proof of the theorem is based on the methods developed by H. Stahl [59]-[63], A. A. Gonchar and the author [27], [55]. These methods obtained some further generalization in the paper in application to systems of polynomials defined by systems of complex orthogonality relations.Together with the characterization of the limit zero distributions of Hermite-Padé polynomials by a vector equilibrium problem we consider an alternative characterization using a Riemann surface R( f ) associated with f . In this terms we present a more general (without Angelesco condition) conjecture on the zero distribution of Hermite-Padé polynomials.Bibliography: 72 items.

On a New Approach to the Problem of Distribution of Zeros of Hermite—Padé Polynomials for a Nikishin System

“…. , f m was also considered in the paper [17], in which, in particular, the effect of pushing of the support of the equilibrium measure inside the original orthogonality interval was discovered; see also [42]). Within the framework of this vector approach, the answer for a pair of functions (1) is 1 Similarly to the way the strong asymptotics of Padé polynomials is described in terms related to the two-sheeted Riemann surface associated (in accordance with the Stahl theory) with an arbitrary function from the class A • (Σ); see [37], [5], [31].…”

Structure of the Nuttall partition for some class of four-sheeted Riemann surfaces

“…(z − v j ), v j are the Chebotarev points of the compact set S. 10 For the definition of the S-property, see [10], [11], [20], [22] and there references given there.…”

Scalar equilibrium problem and the limit distribution of the zeros of Hermite-Padé polynomials of type II

“…Proof. Lemma 1 is proved by the GRS-method (see [8], [25], [28]). As usual, when applying the GRS-method 8 , we assume the contrary, i.e.,…”

“…Note that, under the hypotheses of Theorem 2 and Lemma 1, the GRS-method is much easier to deal with, because the S-compact set F is a finite union of closed intervals of the real line and σ is a positive measure on F ; cf [8],[24],[25],[28]…”